Anton Bovier Lectures on Metastabilitylelievre/Journees_MAS/AB.pdfLarge deviations Wentzell-Freidlin...

Metastability in Stochastic Dynamics, Paris

Anton Bovier

Lectures on Metastability

Based on collaborations with:M. Eckhoff, Veronique Gayrard, M. Klein, A. Bianchi, D. Ioffe, and many others....;

IAM [email protected] 21.09. 2011

Plan

1.Metastability: basic ideas and approaches2.Metastability: potential theoretic approach3.The random field Curie Weiss model

Metastability in Stochastic Dynamics, Paris 21.09. 2011 2(nn)

Metastability: basic ideas

Metastability is a common phenomenon in non-linear dynamics. In particular, itis related to the dynamics of first order phase transitions:

gas

T

P

liquid

Thermally activated transitions between comformations

If the parameters of a systems are changed rapidly across the line of a firstorder phase transition, the system will persist for a long time in a metastablestate before transiting rapidly to the new equilibrium state under the influence ofrandom fluctuations.

Real life examples are numerous:Conformational transitions in molecules, catalytic chemical reactions, global cli-mate changes, market crashes, etc.....


Metastability: phenomenological description

Two main characteristics of metastability are:

⊲Quasi-invariant subspaces Si and

⊲multiple, separated time scales:

⊲Local equilibrium (=’metastable state’) Si reached on a fast scale.

⊲Transitions between Si and Sj occur on much longer metastable time scales.

⊲Due to speparation of time-scales, exponential law of metastable exit times.


Stochastic models

Common context: Markov Processes

Main examples:

⊲Markov chains with finite (or countable) state space.

⊲The classical example of a diffusion process with gradient drift on Ω ⊂ Rd,

e.g. the stochastic differential equation

dXǫ(t) = −∇F (Xǫ(t))dt +√2ǫdW (t)

⊲Stochastic dynamics of lattice spin systems/lattice gases


The canonical model

1940’s: Kramers and Eyring: reaction path theory:

A transition from a metastable is described by themotion of the system along a canonical reaction-path in a free-energy landscape under the drivingforce of noise:

dXt = F ′(Xt)dt +√2ǫdBt


The double well

Kramers one-dimensional diffusion model al-lows quantitative computations: E.g., Kramersobtained the Eyring-Kramers formula for themean transition time:

Eτ =2π√

−F ′′(z∗)F ′′(x)e[F (z∗)−F (x)]/ǫ

f

x

z∗

Further features:⊲Arrival in stable state after many unsuccessful trials;⊲Time of last exit very short compared to Eτ ;⊲τ/Eτ exponentially distributed

0 100000 200000-2

-1

0

1

2

System Metastability

Sigma 0.25 0.2Some big questions remain:⊲What is F for a specific system?⊲How can the simple 1d sde model be justified in examples?


Freidlin-Wentzell theory

The most influential work was the study by MarkI. Freidlin and Alexander D. Wentzell on the the-ory of Small Random Perturbations of DynamicalSystems. This is concerned with generalisationsof Kramers equation to higher dimension, namelySDE’s

dXt = g(Xt)dt +√2ǫ(h(Xt), dBt)

where the unperturbed system, ddtXt = g(Xt) has

multiple attractors.


Large deviations

Wentzell-Freidlin theory is based on the theory of large deviations on path space.

It expresses the probability for a solution, X ǫ, tobe in the vicinity of an specified path, γ : a → b, interm of an action functional

P [‖X ǫ − γ‖∞ ≤ δ] ∼ exp

(−1

ǫ

∫ T

0

L(γ(t), γ′(t))dt

)

This allows to estimate the success probability foran escape from one attractor to another on theleading exponential scale.

This allows the reduction of the analysis of the metastable behaviour of thesystem to that of a Markov chain with exponentially small transition probabilities.

The drawback of this approach are the rather imprecise exponential estimates,that miss, e.g., the prefactor in the Eyring-Kramers formula.


Spectral approach

In the context of diffusion models (multidimensional Kramers equation), analyticmethods from quantum mechanics have been used to compute asymptotic ex-pansions in the diffusivity (ǫ) (Matkovski-Schuss, Buslov-Makarov, Maier-Stein).

-2

-1

0

1

2-2

-1

0

1

2

-15

-10

-5

0

−L = −ǫ∆+∇F · ∇

⇔ -2

-1

0

1

2-2

-1

0

1

2

0

20

40

H = −ǫ2∆ +1

4‖∇F‖2 − ǫ

2∆F

Rigorous analysis of eigenvalues and eigenfunctions via Witten complex byKlein, Helffer, Nier [2005] (aee also Matthieu (95), Miclo (95)).

• Strong point: in principle gives full asymptotic expansions in ǫ.• Drawback: so far only applicable for reversible diffusion processes.


Our approach

Main objectives:

⊲Model independent structural results⊲Link properties of the invariant measure to those of dynamics⊲Good (precise!) calculability of model dependent quantities

This has led us to

⊲Systematic use of potential theoretic ideas⊲Focus on capacities⊲Systematic use of variational principles

Limitations:

⊲So far useful only in reversible Markov processes


Definition of metastability

A key feature in our approach is to representquasi-invariant subsets by a single point (orsmall ball, Bi), and to consider the transitionsbetween them.

A good definition should characterize the travel times,τB ≡ inft > 0 : X(t) ∈ B

A family of Markov processes is metastable, if thereexists a collection of disjoint sets Bi, such that

supx 6∈∪iBiExτ∪iBi

infi infx∈BiExτ∪k 6=iBk

= o(1)

but involve only well-computable quantities.


Martingale and Dirichlet problems

Markov processes are conveniently characterised as solutions to a martingaleproblem associated to a closed, linear dissipative operator, Λ, called generator.

Martingale problem:A stochastic process, Xt, is a Markov process, if for a (suffi-ciently large) class of functions, f ,

f(Xt)− f(X0)−∫ t

0

Lf(Xs)ds

is a martingale. [In discrete time, replace the sum by an integral].

Dirichlet problem: Let L be a dissipative linear operator on S, and D ⊂ S. Letg, k, u : S → R be continuous functions. Can we find a continuous functionf : S → R, such that

−(Lf)(x) + k(x)f(x) = g(x),∀x ∈ D

f(x) = u(x),∀x ∈ Dc.


Martingale and Dirichlet problem

Under suitable conditions, the martingale problem provides a stochastic solutionof the Dirichlet problem:

Stochastic solution: Let τDc ≡ inft > 0 : Xt 6∈ D. If

ExτDce− infx∈D(k(x))τDc < ∞, ∀x ∈ D,

then

f(x) = Ex

[u(XτDc) exp

(−∫ τDc

0

k(Xs)ds

)

+

∫ τDc

0

g(Xs) exp

(−∫ t

0

k(Xs)ds

)dt

]


Representation for probabilitic quantities

Equilibrium potential: Let A,B ⊂ S disjoint then

hA,B(x) ≡ Px (τA < τB)

for x 6∈ A ∪B, solves

LhA,B(x) = 0, x ∈ S \ (A ∪B),

hA,B(x) = 1, x ∈ A,

hA,B(x) = 0, x ∈ B

Mean entrance time: For B ⊂ S,

w(x) ≡ ExτB

solves

−(LwB)(x) = 1, x 6∈ B,

wB(x) = 0, x ∈ B.


Potential theory

Equilibrium measure:

eA,B(dx) ≡ −LhA,B(x), x ∈ A

orhA,B(x) =

∫

A

GB(x, y)eA,B(dy)

with GB Green function.

Except in very simple cases there are no easy solutions of the Dirichlet problem.So why is this useful?


Reversible processes

A Markov process is called reversible with respect to a measure µ, if its genera-tor, L, is self-adjoint on the space L2(S, µ).This entails a number of remarkably powerful facts.

Dirichlet form:

E(f, g) ≡ −∫

S

µ(dx)f(x)(Lg)(x)

positive quadratic form.Capacity:

cap(A,B) ≡∫

A

µ(dx)

dxeA,B(dx) = E (hA,B, hA,B)

For A,B disjoint,

∫

A

νA,B(dz)EzτB =1

cap(A,B)

∫µ(dy)hA,B(y)


Variational principles and monotonicity

Dirichlet priciple:

cap(A,B) = infh∈HA,B

E(h, h)

HA,B = h : h(x) = 1, x ∈ A, h(x) = 0, x ∈ B

⇒ upper bounds!

Monotonicity:

Since the Dirichlet form can be written as an integral (sum) over positive terms,lower bounds can be obtained by removing terms


Lower bounds: Diffusions

Diffusions: Here L = −ǫ∆+∇F (x)∇,

E(h, h) = ǫ

2

∫e−F (x)/ǫ (∇h(x),∇h(x)) dx

Lower bound by reducing integration domain to tube and gradient by directionalderivative

E(h, h) ≥ ǫ

2

∫

√e

dx|

∫ 1

−1

e−F (γ(t)+x|)/ǫ(d

dth(γ(t) + x|)

)2


Example: discrete chains

In discrete state space, the Dirichlet form can be written as

E(f, g) = 1

2

∑

x,y

µ(x)p(x, y) (f(x)− f(y)) (g(x)− g(y))

Best lower bounds use a variational principle from Berman and Konsowa [1990]:

Let f : E → R+ be a non-negative unit flow from A → B, i.e. a function on edgessuch that

⊲∑

a∈A∑

b f(a, b) = 1

⊲ for any a,∑

b f(b, a) =∑

b f(a, b) (Kirchhoff’s law).

Set qf(a, b) ≡ f(a,b)∑b f(a,b)

, and let the initial distribution for a ∈ A be

F (a) ≡∑b f(a, b).

This defines a Markov chain on paths X : A → B, with law Pf .


Examples: discrete state space

Theorem 1. For any non-negative unit flow, f , one has that, for X =(a0, a1, . . . , a|X |),

cap(A,B) = supf :A→B

= EfX

|X |−1∑

ℓ=0

f(aℓ, aℓ+1)

µ(aℓ)p(aℓ, aℓ+1)

−1

Equality is reached for the harmonic flow

f(a, b) =1

cap(A,B)µ(a)p(a, b) [hH,B(b)− hA,B(a)]+


Finite state Markov chains

In the simplest (but already rich) setting of finite state Markov chain, applicationof potential theory yields a very clear and simple picture.

Definition

A Markov processes Xt is ρ-metastable with respect tothe set of metastable points M ⊂ S, if

supx∈M Px[τM\x < τx]

infz 6∈M Pz[τM < τz]≤ ρ ≪ |S|−1

⇔infx∈M cap(x,M\ x)/µ(x)supz 6∈M cap(z,M)/µ(z)

≤ ρ ≪ |S|−1

Note: we will always think of a sequence of processes Xn, where ρ = ρn, but alsopossibly S = Sn; our definition says that ρn|Sn| → 0.


Consequences of the definition

Basins of Attraction: x ∈ M,

A(x) = y ∈ Γ : Py(τx = τM) = maxz∈M

Py(τz = τM)

Metastable exits: ExτMx wherex ∈ M and Mx are all y ∈ Ms.t. µ(y) ≥ µ(x).

A(x)

x

Mx


Theorem Exit times and spectrum

Theorem 1:[BEGK ’02] If a reversible Markov process with finite state space ismetastable with metastable set M, then:

ExτMx =µ(A(x))

cap(x,Mx)(1 + o(1))

andPx[τMx/ExτMx > t] = (1 + o(1))e−t(1+o(1))

For each x ∈ M there exists one eigenvalue, λx, of L, s.t.

λx =1

ExτMx

(1 + o(1))

The corresponding right-eigenfunctions, φi, satisfy s.t.

φx(y) ∼ Py(τx < τMx) + o(1)

[all under some non-degeneracy hypothesis]


Tools

The magic formula:

ExτB =1

cap(x,B)

∑

y∈Sµ(y)hx,A(y)

Renewal inequality:

hA,B(x) ≤cap(x,B)

cap(x,A)

Ultrametric triangle inequality: For any x, y, z,

cap(x, y) ≥ 1

3min (cap(x, z), cap(y, z))

Variational representations to estimate capacities.


Eigenvalues.

The connection between metastable behaviour and the existence of small eigen-values of the generator of the Markov process has been realised for a very longtime. Some key references are Davies, Wentzell, Freidlin-Wentzell, Mathieu,Miclo, Gaveau-Schulman, etc.

The key idea of our approach is the observation that if fλ is an eigenfunction ofL, then it must be representable as the solution of a boundary value problemwith suitable boundary conditions on M:

(L− λ)f(y) = 0, y ∈ Ω \M; f(x) = fλ(x), x ∈ M (∗)This allows to characterize all eigenvalues that are smaller than the smallesteigenvalue, λ0, of the Dirichlet operator in Ω \M.A priori estimates. The first step consists in showing that the matrix LM withDirichlet conditions in all the points in M is not very small:

Lemma: (Donsker-Varadhan) Let λ0 denote the infimum of the spectrum of−LM. Then

λ0 ≥ 1

supx∈ΩEzτM


Ideas of the proof. Eigenvalues.

Let us denote by EM(λ) the |M| × |M|- matrix with elements

(EM(λ))xy ≡ eλz,M\z(x)

Lemma A number λ < λ0 is an eigenvalue of the matrix L if and only if

det EM(λ) = 0

Anticipating that we are interested in small λ, we want can use perturbationtheory to find the roots of this non-linear characteristic equations.


Beyond finite state space

The conditon ρn|Sn|| ↓ 0 fails when:

⊲The state space is uncountable[Diffusions]

⊲Cardinality of state space grows too fast[lattice spin systems at finite temperature]

Reason in all cases is that processes will not hit single points fast enough.

Remedy: Replace points in definition of metastability be small balls.

Difficulty: Complicates all renewal arguments!

Needed: A priori control of regularity of harmonic functions and other solutions ofDirichlet problems.


Trivial examples: Mean field models

State space: Sn = −1, 1n Hamiltonian:

Hn(σ) = − 1

2n

n∑

i,j=1

σiσj =n

2(mn(σ))

2

where mn(σ) =1n

∑ni=1 σi.

Rates:

pn(σ, σ′) = exp

(− [Hn(σ

′)−Hn(σ)]+)

mλ(σ(t)) again Markov chain on −1,−1− 2/N, . . . , 1;

⊲nearest neigbor random walk reversible with respect to measureexp(−βNF (x)) with F free energy;

⊲explicitely solvable;

Thus here we essentially have exactly the situation imagined by Kramers andEyring.


The real thing:

Whenever we are not in one of the two situations above, we have problems:

⊲There is no exact reduction to a finite dimensional system!

Still, we expect an effective description of the dynamics in terms of some meso-scopic coarse grained dynamics!

In the remainder of this talk I will explain how this idea can be implemented in asimple example.


The RFCW model

Random Hamiltonian:

HN(σ) ≡ −N

2

(1

N

N∑

i=1

σi

)2

−N∑

i=1

hiσi.

hi, i ∈ N are (bounded) i.i.d. random variables, σ ∈ −1, 1N .

Equilibrium properties: [see Amaro de Matos, Patrick, Zagrebnov (92), Kulske (97)]

Gibbs measure: µβ,N(σ) =2−Ne−βHN (σ)

Zβ,N

Magnetization: mN(σ) ≡ 1N

∑Ni=1 σi.

Induced measure: Qβ,N ≡ µβ,N m−1N . on the set ΓN ≡ −1,−1 + 2/N, . . . ,+1.


Equilibrium properties

Using sharp large deviation estimates, one gets

Zβ,NQβ,N(m) =

√2I ′′N(m)

Nπexp −NβFN(x) (1 + o(1)) ,

where FN(x) ≡ 12m

2 − 1βIN(m) and IN(y) is the Legendre-Fenchel transform of

UN(t) ≡1

N

∑

i∈Λln cosh (t + βhi)

Critical points: Solutions of m∗ = 1N

∑i∈Λ tanh(β(m

∗ + hi)).Maxima if βEh

(1− tanh2(β(z∗ + h))

)> 1.

Moreover, at critical points,

Zβ,NQβ,N(z∗) =

expβN

(−1

2 (z∗)2 + 1

βN

∑i∈Λ ln cosh (β(z

∗ + hi)))

√Nπ2

(Eh

(1− tanh2(β(z∗ + h))

)) (1 + o(1))

-1.0 -0.5 0.5 1.0

-0.482

-0.481

-0.480

-0.479

-0.478


Glauber dynamics

We consider for definiteness discrete time Glauber dynamics with Metropolistransition probabilities

pN(σ, σ′) ≡ 1

Nexp −β[HN(σ

′)−HN(σ)]+

if σ and σ′ differ on a single coordinate, and zero else.

We will be interested in transition times from a local minimum, m∗, to the set of“deeper” local minima,

M ≡ m : Fβ,N(m) ≤ Fβ,N(m∗).

Set S[M ] = σ ∈ SN : mN(σ) ∈ M.

We need to define probability measures on S[m∗] by

νm∗,M(σ) =µβ,N(σ)Pσ

[τS[M ] < τS[m∗]

]∑

σ∈S[m∗] µβ,N(σ)Pσ

[τS[M ] < τS[m∗]

].


Main theorem

Theorem 2. Let m∗ be a local minimum of Fβ,N ; let z∗ be the critical pointseparating m∗ from M .

Eνm∗,MτS[M ] = exp βN (FN(m∗)− Fn(z

∗))

× 2πN

β|γ1|

√βEh

(1− tanh2 (β(z∗ + h))

)− 1

1− βEh

(1− tanh2 (β(m∗ + h))

) (1 + o(1)) ,

where γ1 is the unique negative solution of the equation

Eh

[1− tanh(β(z∗ + h))

[β (1 + tanh(β(z∗ + h)))]−1 − γ

]= 1.

Note that a naive approximation by a one-dimensional chain would give thesame result except the wrong constant

γ =1

βEh

(1− tanh2 (β(z∗ + h))

) − 1


Previous work

The model was studied in⊲F. den Hollander and P. dai Pra (JSP 1996) [large deviations, logarithmic

asymptotics]⊲P. Mathieu and P. Picco (JSP, 1998) [binary distribution; up to polynomial er-

rors in N ]⊲A.B, M. Eckhoff, V. Gayrard, M. Klein (PTRF, 2001) [discrete distribution, up

to multiplicative constants]

Both MP and BEGK made heavy use of exact mapping to finite-dimensionalMarkov chain!

The main goal of the present work was to show that potential theoretic methodsallow to get sharp estimates (i.e. precise pre-factors of exponential rates) inspin systems at finite temperature when no symmetries are present. The RFCWmodel is the simplest model of this kind.


Remark on starting measure

The discusion above explains why it is natural in our formalism to get results forhitting times of the process started in the special measure νm∗.

Of course one would expect that in most cases, the same results hold unifomrlypointwise within a suitable set of in itial configurations, i.e. Eνm∗τS[M ] ∼ EστS[M ],for all σ ∈ S[m∗].

In our case, we can show this to be true using a rather elaborate coupling argu-ment (BiBoIo, 2009). This allows also to prove that τS[M ]/Eνm∗τS[M ] is asymptoti-

cally exponentially distributed.


Coarse graining

Iℓ, ℓ ∈ 1, . . . , n: partition of the support of the distribution of the random field.

Random partition of the set Λ ≡ 1, . . . , NΛk ≡ i ∈ Λ : hi ∈ Ik

Order parameters

mk(σ) ≡1

N

∑

i∈Λk

σi

HN(σ) = −NE(m(σ)) +

n∑

ℓ=1

∑

i∈Iℓ

σihi

where hi = hi − hℓ, i ∈ Λℓ. Note |hi| ≤ c/n;

E(x) ≡ 1

2

(n∑

ℓ=1

xℓ

)2

+n∑

ℓ=1

hℓxℓ

Equilibrium distribution of the variables m[σ]

µβ,N(m(σ) = x) ≡ Qβ,N(x)


Elements of Proof 2: Coarse graining

Coarse grained Dirichlet form:

Φ(g) ≡∑

x,x′∈ΓN

Qβ,N [ω](x)rN(x,x′) [g(x)− g(x′)]

2

withrN(x,x

′) ≡ 1

Qβ,N [ω](x)

∑

σ:m(σ)=x

µβ,N [ω](σ)∑

σ′:m(σ)=x′p(σ, σ′).


Approximate harmonic functions

The key step in the proof of both upper and lower bounds is to find a functionthat is almost harmonic in a small neighborhood of the relevant saddle point.This will be given by

h(σ) = g(m(σ)) = f((v, (z∗ −m(σ))))

for suitable vector v ∈ Rn and f : R → R+

f(a) =

√βN γ

(n)1

2π

∫ a

−∞e−βN |γ1|u2/2du.

This yields a straightforward upper bound for capacities which will turn out to bethe correct answer, as n ↑ ∞!


Construction of flow

Lower bound on capacity uses the Berman-Konsowa variational principle.Again, care has to be taken in the construction of the flow only near the saddlepoint.

Two scale construction:

⊲Construct mesoscopic flow on variables m from approximate harmonic func-tion used in upper bound. This gives good lower bound in the mesoscopicDirichlet form.

⊲Construct microscopic flow for each mesoscopic path.

⊲Use the magnetic field is almost constant and averaging that conductanceof most mesoscopic paths give the same values as in mesoscopic Dirichletfunction.

This yields upper lower bound that differs from upper bound only by factor 1 +O(1/n)).


Result for capacity

If A = σ : mN(σ) = m1, B = σ : mN(σ) = m2, and z∗ is the essential saddle

point connecting them, then

cap(A,B) = Qβ,N(z∗)β|γ(n)

1 |2πN

(n∏

ℓ=1

√rℓ

)(πN

2β

)n/21√∏nj=1 |γj|

(1 +O(ǫ))

This can be re-written as:

Theorem 3.

Zβ,Ncap(A,B) =β|γ(n)

1 |2πN

expβN

(−1

2FN(z∗))

(1 + o(ǫ))√βEh

(1− tanh2(β(z∗ + h))

)− 1

.


Control of harmonic function

Final step in control of mean hitting times:Compute∑

σ

µβ,N(σ)hA,B(σ) ∼ Qβ,N([ρ +m1, m1 − ρ])

This requires to show that: hA,B(σ) ∼ 1, if σnear A, andhA,B(σ) ≤ exp −N(Fβ,N(z

∗)− Fβ,N(mN(σ))− δ)if Fβ,N(mN(σ) ≤ Fb,N(m1).

m1 m2z∗

h ∼ 1

h ∼ 0

h ≤ 1

Can be done using super-harmonic barrier function.


Future Challenges

Some further problems that have been or are being treated:⊲Discrete diffusions (BEGK ´00)⊲Glauber dynamics for Ising models, T ↓ 0 (B, F. Manzo ’01)⊲Kawasaki dynamics, T ↓ 0, (B,den Hollander, Nardi 05, B, dH, Spitoni 10)⊲SPDE in d = 1 (Barret, B, Meleard 10, Barret 11)⊲Ageing in disordered systems :•Random Energy model (G. Ben Arous, B, V. Gayrard ’03)•REM - like trap model (B., A. Faggionato ’05)•Sinai’s random walk (B, A. Faggionato, ’08)

Major challenges:⊲Extension of the general theory to infinite dimensional cases:⊲Lattice models (Glauber, Kawasaki)•at finite temperatures• in infinite volume⊲ Infinite dimensional diffusion processes in d > 1?


References: Metastability

⊲A. Bovier, M. Eckhoff, V. Gayrard, and M. Klein, “Metastability and low-lyingspectra in reversible Markov chains”, CMP 228, 219-255 (2002).

⊲A. Bovier, M. Eckhoff, V. Gayrard, and M. Klein, “Metastability in reversiblediffusion processes I. Sharp asymptotics for capacities and exit times”, JEMS6, 399-424 (2004).

⊲A. Bovier, V. Gayrard, and M. Klein, “Metastability in reversible diffusion pro-cesses II. precise asymptotics for small eigenvalues”, JEMS 7 69–99 (2005).

⊲A. Bovier, F. den Hollander and C. Spitoni, “Homogeneous nucleation forGlauber and Kawasaki dynamics in large volumes and low temperature”, Ann.Probab. 38, 661–713 (2010).

⊲A. Bianchi, A. Bovier, and D. Ioffe, “Sharp asymptotics for metastability in theRandom Field Curie-Weiss model”, EJP 14, 1541-1603 (2009),

⊲A. Bianchi, A. Bovier, and D. Ioffe, “Pointwise estimates and exponential lawsin metatable systems via coupling methods”, Ann. Probab., (2011).

⊲F. Barret, A. Bovier, and S. Meleard, Uniform estimates for metastable transi-tion times in a coupled bistable system, EJP 15, 323–345 (2010).

⊲A. Bovier “Metastability”, in Methods of Contemporary Mathematical Statisti-cal Mechanics (R. Kotecky, ed.) Springer LNM 1970, 2009.


References: Ageing

⊲G. Ben Arous, A. Bovier, and V. Gayrard, “Glauber dynamics of the randomenergy model. 1. Metastable motion on the extreme states.”, Commun. Math.Phys. 235, 379-425 (2003).

⊲G. Ben Arous, A. Bovier, and V. Gayrard, “Glauber dynamics of the randomenergy model. 2. Aging below the critical temperature”, Commun. Math.Phys. 236, 1-54 (2003).

⊲G. Ben Arous A. Bovier, and V. Gayrard, “Aging in the random energy model”,Phys. Rev. Letts. 88, 087201 (2002).

⊲A. Bovier and A. Faggionato “Spectral chacterisation of ageing: the REM-liketrap model”, Ann. Appl. Prob. 15, 1997–2037 (2005).

⊲A. Bovier and A. Faggionato, “Spectral analysis of Sinai’s walk for small eigen-values”, WIAS-preprint, submitted to Ann. Probab. (2005).


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